Math 2.0

statement proposition

A series of nonnegative terms converges iff its partial sums are bounded

Statement

Let (an)(a_n) be a real sequence with an≥0a_n \ge 0 for all nn. Then the series ∑an\sum a_n converges if and only if its partial sums sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n are bounded above. In this case, ∑n=0∞an=sup⁡NsN\sum_{n=0}^{\infty} a_n = \sup_N s_N.

Remarks

The partial sums increase, since sN+1−sN=aN+1≥0s_{N+1} - s_N = a_{N+1} \ge 0. The result is then the monotone convergence theorem. If the partial sums are not bounded, they tend to +∞+\infty, and one writes ∑an=+∞\sum a_n = +\infty. The comparison test, the integral test and the Cauchy condensation test all rest on this fact.

Explanations

L1 human

When all terms are nonnegative, the running total can only go up. So it either stays below some ceiling and converges, or it grows past every bound.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Partial sum definition
  10. Convergent series definition
  11. A series of nonnegative terms converges iff its partial sums are bounded statement

What depends on this

93 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Monotone_convergence_theorem, Convergence of a monotone series

What links here

Used by